The Poisson regression with fixed and random effects in non-life insurance ratemaking

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1 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna Th Posson rgrsson wth fxd and random ffcts n non-lf nsranc ratmakng Alcja Wolny-Domnak 1, Danl Sobck Abstract Important part of data analyss n nsranc bsnss s th constrcton of a far tarff strctr calld classfcaton ratmakng. Th goal of ths classfcaton s partton all polcs n partclar portfolo nto homognos classs. Wthn vry class, all polcyholdrs pay th sam pr prmm. To dsgn classfcaton ratng plans, actars s th gnralzd lnar modls (GLM tchnq. In GLM modl, th dpndnt varabl s sally th clam svrty or th clam frqncy for th polcy. In th papr w focs on th clam frqncy. Th ratng varabls ar th catgorcal varabls wth fw catgors lk.g. gndr or a larg nmbr of catgors lk.g. spatal varabls. Th GLM modl assms obsrvd rsponss ar ndpndnt. Howvr many portfolos of polcs yld corrlatd obsrvatons. Corrlaton rslts from th samplng dsgn or th way data ar collctd. Th am of ths papr s to propos mxd modls basd on Posson rgrsson whch ar sfl n clam frqncy modllng. In ths modls w tak nto accont spcfcts of nsranc data: corrlaton, ovrdsprson and zro-nflaton ffcts n data. Th cas stdy dmonstrat th valdty of th applcaton of ths modls. Kywords: clam frqncy, mxd modl, Posson rgrsson, ZIP, ZIGP JEL Classfcaton: C1, C51 1. Introdcton Modllng clam frqncy n th portfolo of polcs s an ssntal part of non-lf nsranc ratmakng n th portfolos of polcs. Th ratmakng s dfnd as rsk classfcaton, whch nvolvs th gropng polcs nto varos classs that shar a homognos st of charactrstcs nflncs clam frqncy. In vry class th sam nt prmm, calclatd as th xpctd clam frqncy n ths cas, s than rasonabl. Th ratmakng s sally don n two stps (Dnt, Maréchal t al. 007; Bochr and Gllén, 009. In th frst stp, calld a pror ratmakng, polcs n th portfolo ar classfd accordng masrabl nformaton abot th polcyholdr and nsrd objct (Antono and Brlant, 006; d Jong and Hllr, 008; Wolny-Domnak and Stdnk, 013. Aftr a pror classfcaton, th portfolo s dvdd nto homognos grops, bt only of th obsrvabl factors. Som mportant hddn charactrstcs stll gnrat htrognty n vry grop of polcs (.g. n 1 Unvrsty of Economc n Katowc, Dpartmnt of Statstcal and Mathmatcal Mthods n Economcs, 1 Maja 50, Katowc, Poland; alcja.wolny-domnak@.katowc.pl Warsaw School of Economcs, Collgm of Economc Analyss, Madalńskgo 6/8, Warsaw; sobckd@gmal.com 3

2 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna atomobl nsranc th bhavoral charactrstcs of drvr. That s why th hstory of clams as t mrg for ndvdal polcy s takn nto accont n scond stp of ratmakng, calld a postror ratmakng (Antono and Valdz, 01. In a pror ratmakng crosssctonal data ar sd whl n a postror ratmakng rathr longtdnal strctr. As th clam frqncy s an xampl of cont data thr ar fw problms n modllng of sch a data. Ltratr rvw rvals that, n partclar, attmpts ar ndrtakn to fnd a probablstc modl for th clam frqncy dstrbton, whr sally ths dstrbton s assmd to b Posson. Howvr th nsranc portfolos hav a vry spcfc charactrstc,.. for many polcs thr ar no clams obsrvd n th nsranc hstory for a gvn prod of tm. It mans that th data contans lots of zros and, as a consqnc, th Posson rgrsson may not gv satsfactory rslts (zro-nflaton ffct. In ordr to allow th prsnc of xcss zros n nsranc portfolo, th zro-nflatd modls ar appld (Lambrt, 199; Yp and Ya, 005; Wolny-Domnak, 013. Th classc modl s th zro nflatd Posson modl (ZIP, whch s a mxtr of a Posson dstrbton and a zro pont mass. Th othr problm oftn xstng n nsranc data s th ncdnc of ovrdsprson, whch mans that data xhbt gratr varablty than allowd to th Posson modl and th man s not qal to varanc (ovrdsprson ffct. Th rason of that may b th dsrgardng som latnt factors affctng th clams occrrnc. Th gnralzaton of th Posson modl s possbl and than th gnralzd Posson modl (GP s rcvd (Consl and Famoy, 199. Th gnralzd Posson dstrbton sally s sd whn th occrrnc of clams s probably dpndnt, whch s a common staton n non-lf nsranc (Yp and Ya, 005. Usally n cas of ovrdsprson n ZIP modl, zro-nflatd ngatv bnomal (ZINB modl s sd (Hall, 000, bt th zro-nflatd gnralzd Posson (ZIGP s also possbl. In ltratr thr ar som smlaton stds wth th scor tst for ovrdsprson basd on ZIGP modl, whch llstrat that ZIGP modl has hghr mprcal powr than ZINB modl (Yang t al., 009. Wthn th contxt of a pror ratmakng, t s bcomng a standard norm n practc to s Gnralzd Lnar Modls (GLMs whr cross-sctonal data s modlld wthn th class of xponntal dsprson dstrbtons (Habrman and Rnshaw, 1996; Ohlsson, 008. Th GLM modl assms obsrvd rsponss ar ndpndnt. Howvr many portfolos of polcs yld corrlatd obsrvatons. Corrlaton rslts from th samplng dsgn or th way data ar collctd. Th corrlatd data typcally has a clstrd strctr,.g. th atomobl portfolo wth th spatal varabl lk gographcal rgons. In ths cas th sorc of corrlaton cold b xplan n followng way: ach rgon s lkly to xprnc roghly th sam wathr condtons and hnc dffrnt polcs n th sam rgon ar lkly to hav a 33

3 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna smlar clams xprnc. Smlarly for a postror ratmakng, as n longtdnal stds, whr th obsrvatons rprsnt rpatd otcoms from ndvdal sbjcts, th corrlaton rspons at on tm s corrlatd wth th rspons at anothr tm. Th longtdnal data s a spcal cas of clstrd data n whch th clstr s th polcyholdr. Ignorng th corrlaton can lad to rronos conclsons (d Jong and Hllr, 008. That s why w propos approprat mxd modls basd on Posson mxd rgrsson modls n ratmakng. In th papr w propos two modls basd on th mxd Posson rgrsson modl, whch can b sd n a pror ratmakng as wll as a postror. In th cas stdy w consdr th ral portfolo of atomobl polcs takn from polsh nsranc company. As th portfolo contans th cross-sctonal data strctr w analyz zro-nflaton and ovrdsprson ffcts n th portfolo gvn th spatal varabl as random ffct. Th stmaton s don wth margnal lklhood mthod (MLM sng NLMIXED procdr from SAS (th ntgraton mthod s Adaptv Gassan Qadratr (s Lttll, Th mxd modls basd on Posson rgrsson Assm th portfolo of n polcs. Lt N dnots th clam frqncy for ndvdal polcy. Sppos jth rspons varabl from th clstr follows a Posson dstrbton: Pr POIS n n], (1 n! 1,..., p and j 1,...,n, whr p s th nmbr of clstrs and n s th nmbr of obsrvatons wthn clstr. In cas of th ovrdsprson ffct, th probablty dnsty fncton (1 shold b xtndd to th gnralzd Posson dstrbton gvn by: Pr GP n n] ( 1 (1 n n 1 n! (1 n xp[ 1 ] ( wth th man and th varanc rspctvly: E(, N Var ( N (1. If 0, th abov modl rdcs to th Posson dstrbton wth no ovrdsprson ffct. In th staton of xcss zros n th portfolo, th ZI-probablty dnsty fncton s rcommndd to s. Th gnral form of Pr n ] can b xprss as follows: ZI Pr ZI n (1 Pr n], ] (1 Pr[ N n], n 0, (3 n 0 34

4 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna whr s th probablty of zro clam frqncy for th polcy n j th clstr and Pr n ] s th probablty dnsty fncton (1 or (. Th zros from frst qaton ar calld strctral zros and from scond qaton samplng zros. Frst two momnts n zro - nflatd modls ar: for th ZIP dstrbton and N for th ZIGP dstrbton. E[ ] (1, Var[ ] (1 ( (4 N N E[ ] (1, Var[ N ] E[ ][(1 ] (5 N In ratmakng w ar ntrstd n xtndd probablty dnsty fnctons (1 (3 to modls wth xplanatory varabls. In th rgrsson sttng, th man, zro proporton and ar rlatd to th covarats vctors x, z and w rspctvly. Rsponss wthn th sam clstr ar lkly to b corrlatd. To accommodat th nhrnt corrlaton, random ffcts ar ncorporatd n th lnar prdctors x β. Gvn th vctor of random ffcts, w propos followng modls: 1. Th mxd Posson rgrsson wth random ntrcpt (dnotng by POIS-M: ( 1,..., p N ~ Pos ( x (, X1,..., X k β β, (6 ~ N(0,. Th mxd Posson rgrsson whn zp-nflaton occrs (dnotng by GP-M: N ~ ZIP(, x β ( β, X1,..., X p T z ( γ, Z1,..., Zq 1 ~ N(0, γ z T γ 1 1 T z γ, (7 35

5 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna 3. Th mxd Posson rgrsson whn ovrdsprson and zro-nflaton occr (dnotng by ZIGP-M: N ~ ZIGP(,, x β ( β, X 1,..., X k w (, W1,..., Wt 1 z γ (,,..., γ Z1 Z q z γ 1 ~ N(0, 1 1 z γ, (8 Th lnk fncton n ZIGP-M modl for paramtr ar takn from (Czado t al, 007. Th random ffcts ar assmd to b ndpndnt and normally dstrbtd, ~ N(0,. Basd on th gnralzd lnar mxd modl formlaton (Brslow and Clayton, 1993; McClloch, 006, th margnal lklhood mthod (MLM stmats of (6-(8 mxd rgrsson modls paramtrs can b obtan. Th gnral form of th margnal lklhood can b than xprss as follows: m 1 m n l( log L ( log Pr n ] f ( d, (9 1 j1 whr s th vctor of paramtrs n th modl and f dnots th normal dnsty fncton for th random ffcts. Th fncton L ( s of th form (p to th modl: L ( ZIGP Pr GP L L n ( POIS M (, PrPOIS n] f ( j1 ( ZIP M [(1 Pr ( β,, γ, n ] I ( n 0 0 dla y 0 whr IY 0. 1 dla y 0 β d, (10 ( β, γ, n ZIP j1 n ] f ( d j 1 n ] I [( (1 ( n 0 ] f ( d I ( n 0 [( (1 xp[ ] I 1 ], (11 ( n 0 ][(1, (1 Thr ar fw mthods to stmat abov mxd modls (Wolfngr and Oconnll, 1993; McClloch, 1997; Lttll, 006. Up to th mthod thr s a nd to obtan th margnal lklhoods (10-(1 by ntgratng ot random ffcts. As th analytcal solton of 36

6 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna ntgrals s ntractabl, on frst apply nmrcal approxmaton: Laplac transformaton or Gass-Hrmt qadratr. In clam frqncy modlng w s margnal lklhood mthod (MLM wth NLMIXED procdr from SAS (th ntgraton mthod s Adaptv Gassan Qadratr. 3. Th mxd Posson rgrsson a pror ratmakng W analyz cross-sctonal sampl of th atomobl nsranc portfolo of a company opratng n Poland. Only prvat-s cars ar consdrd n ths sampl. Thr ar 4 catgorcal xognos varabls as wll as th clam frqncy for vry polcy n th portfolo at falt that wr rportd wthn th yarly prod: CLIENT_AGE (th polcyholdr s ag, 6 catgors, CAR_AGE (th car s ag, 3 catgors, POWER (th ngn powr, 3 catgors, VOIVODESHIP (th rgon n Poland, 16 catgors. W modl th varabl CLAIM_COUNT (clam frqncy. Th xognos nformaton s codd by mans of bnary varabls. W consdr th portfolo wth polcs. As th fracton of zros s % n th portfolo (polcs wth no clams w sspct that zro-nflaton and ovrdsprson ffcts appar n data. Consqntly w nvstgat ZIP-M and ZIGP-M modls wth random ffct assmd to b th spatal varabl VOIVODESHIP. In ordr to analyz th valdty of th applcaton of POIS-M, ZIP-M and ZIGP-M w stmat paramtrs of modls for th whol portfolo. Ths allows s to tst statstcal sgnfcanc of paramtrs,,. Th rslts ar shown n Tab. 1 and Tab.. W obsrv that th varanc componnt s statstcally sgnfcant n all thr modls as wll as othr paramtrs: n ZIP-M modl and, n ZIGP-M modl whch that sng mxd modls tak nto consdraton zro-nflaton and ovrdsprson ffcts s rasonabl. In or portfolo th modl ZIP-M aganst ZIGP-M s prfrabl accordng AIC. Paramtr Estmat (s.. POIS-M p-val ( < ( log-lkhood AIC Tabl 1 Paramtr stmats for POIS-M. 37

7 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna Paramtr Estmat (s.. ZIGP-M p-val Estmat (s.. ZIP-M p-val ( < (0.037 < ( < (0.049 < (0.087 < ( ( log-lkhood AIC Tabl Paramtr stmats for ZIGP-M and ZIP-M modls wth no covarats. Conclsons Ratmakng s an xtrmly mportant part of stablshng rasonabl classfcaton for a portfolo of nsranc polcs. In th ltratr thr s a lot of rgrsson modls proposd to b sd n ths problm. W focsd on th modfd mxd Posson modls wth spatal random ffct whch handl wth zro-nflaton and ovrdsprson ffcts. In th cas stdy w nvstgatd th valdty of ths approach by tstng th statstcal sgnfcanc of paramtrs, and. Ths s prlmnary analyss whch s sfl n th fnal slcton of th modl n ratmakng of partclar portfolo of polcs. Acknowldgmnts Th work of Alcja Wolny-Domnak was spportd by grant NN from Polsh Natonal Scnc Cntr. Rfrncs Antono, K., & Valdz, E. A. (01. Statstcal concpts of a pror and a postror rsk classfcaton n nsranc. AStA Advancs n Statstcal Analyss, 96(, Antono, K., & Brlant, J. (006. Rsk Classfcaton n Nonlf Insranc. Encyclopda of Qanttatv Rsk Analyss and Assssmnt. Brslow, N. E., & Clayton, D. G. (1993. Approxmat nfrnc n gnralzd lnar mxd modls. Jornal of th Amrcan Statstcal Assocaton, 88(41, 9-5. Bochr, J. P., & Gllén, M. (009. A srvy on modls for panl cont data wth applcatons to nsranc. RACSAM-Rvsta d la Ral Acadma d Cncas Exactas, Fscas y Natrals. Sr A. Matmatcas, 103(,

8 Procdngs of th 8 th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna Consl, P. C., & Famoy, F. (199. Gnralzd Posson rgrsson modl. Commncatons n Statstcs-Thory and Mthods, 1(1, Czado, C., Erhardt, V., Mn, A., & Wagnr, S. (007. Zro-nflatd gnralzd Posson modls wth rgrsson ffcts on th man, dsprson and zro-nflaton lvl appld to patnt otsorcng rats. Statstcal Modllng, 7(, D Jong, P., & Hllr, G. Z. (008. Gnralzd lnar modls for nsranc data. Cambrdg Unvrsty Prss. Dnt, M., Maréchal, X., Ptrbos, S., & Walhn, J. F. (007. Actaral modllng of clam conts: Rsk classfcaton, crdblty and bons-mals systms. John Wly & Sons. Habrman, S., & Rnshaw, A. E. (1996. Gnralzd lnar modls and actaral scnc. Statstcan, 45(4, Hall, D. B. (000. Zro nflatd Posson and bnomal rgrsson wth random ffcts: a cas stdy. Bomtrcs, 56(4, Lambrt, D. (199. Zro-nflatd Posson rgrsson, wth an applcaton to dfcts n manfactrng. Tchnomtrcs, 34(1, Lttll, R. C. (006. SAS for mxd modls. SAS nsttt. McClloch, C. E. (1997. Maxmm lklhood algorthms for gnralzd lnar mxd modls. Jornal of th Amrcan statstcal Assocaton, 9(437, McClloch, C. E. (006. Gnralzd lnar mxd modls. John Wly & Sons, Ltd. Ohlsson, E. (008. Combnng gnralzd lnar modls and crdblty modls n practc. Scandnavan Actaral Jornal, 008(4, Wolfngr, R., & Oconnll, M. (1993. Gnralzd lnar mxd modls a psdo-lklhood approach. Jornal of statstcal Comptaton and Smlaton, 48(3-4, Wolny-Domnak, A. (013. Zro-nflatd clam cont modlng and tstng a cas stdy. Ekonomtra, 1(39, Wolny-Domnak, A., & Stdnk, J. (013. Estmaton of clam conts qantls. In Papż, M. & Śmch, S. (Eds., Procdngs of 7th Profssor Alksandr Zlas Intrnatonal Confrnc on Modllng and Forcastng of Soco-Economc Phnomna. Cracow Unvrsty of Economcs, Poland, Yang, Z., Hardn, J. W., & Addy, C. L. (009. Tstng ovrdsprson n th zro-nflatd Posson modl. Jornal of Statstcal Plannng and Infrnc, 139(9, Yp, K. C., & Ya, K. K. (005. On modlng clam frqncy data n gnral nsranc wth xtra zros. Insranc: Mathmatcs and Economcs, 36(,